# The Amplitudes Long View

Occasionally, other physicists ask me what the goal of amplitudes research is. What’s it all about?

I want to give my usual answer: we’re calculating scattering amplitudes! We’re trying to compute them more efficiently, taking advantage of simplifications and using a big toolbox of different approaches, and…

Usually by this point in the conversation, it’s clear that this isn’t what they were asking.

When physicists ask me about the goal of amplitudes research, they’ve got a longer view in mind. Maybe they’ve seen a talk by Nima Arkani-Hamed, declaring that spacetime is doomed. Maybe they’ve seen papers arguing that everything we know about quantum field theory can be derived from a few simple rules. Maybe they’ve heard slogans, like “on-shell good, off-shell bad”. Maybe they’ve heard about the conjecture that N=8 supergravity is finite, or maybe they’ve just heard someone praise the field as “demoting the sacred cows like fields, Lagrangians, and gauge symmetry”.

Often, they’ve heard a little bit of all of these. Sometimes they’re excited, sometimes they’re skeptical, but either way, they’re usually more than a little confused. They’re asking how all of these statements fit into a larger story.

The glib answer is that they don’t. Amplitudes has always been a grab-bag of methods: different people with different backgrounds, united by their interest in a particular kind of calculation.

With that said, I think there is a shared philosophy, even if each of us approaches it a little differently. There is an overall principle that unites the amplituhedron and color-kinematics duality, the CHY string and bootstrap methods, BCFW and generalized unitarity.

If I had to describe that principle in one word, I’d call it minimality. Quantum field theory involves hugely complicated mathematical machinery: Lagrangians and path integrals, Feynman diagrams and gauge fixing. At the end of the day, if you want to answer a concrete question, you’re computing a few specific kinds of things: mostly, scattering amplitudes and correlation functions. Amplitudes tries to start from the other end, and ask what outputs of this process are allowed. The idea is to search for something minimal: a few principles that, when applied to a final answer in a particular form, specify it uniquely. The form in question varies: it can be a geometric picture like the amplituhedron, or a string-like worldsheet, or a constructive approach built up from three-particle amplitudes. The goal, in each case, is the same: to skip the usual machinery, and understand the allowed form for the answer.

From this principle, where do the slogans come from? How could minimality replace spacetime, or solve quantum gravity?

It can’t…if we stick to only matching quantum field theory. As long as each calculation matches one someone else could do with known theories, even if we’re more efficient, these minimal descriptions won’t really solve these kinds of big-picture mysteries.

The hope (and for the most part, it’s a long-term hope) is that we can go beyond that. By exploring minimal descriptions, the hope is that we will find not only known theories, but unknown ones as well, theories that weren’t expected in the old understanding of quantum field theory. The amplituhedron doesn’t need space-time, it might lead the way to a theory that doesn’t have space-time. If N=8 supergravity is finite, it could suggest new theories that are finite. The story repeats, with variations, whenever amplitudeologists explore the outlook of our field. If we know the minimal requirements for an amplitude, we could find amplitudes that nobody expected.

I’m not claiming we’re the only field like this: I feel like the conformal bootstrap could tell a similar story. And I’m not saying everyone thinks about our field this way: there’s a lot of deep mathematics in just calculating amplitudes, and it fascinated people long before the field caught on with the Princeton set.

But if you’re asking what the story is for amplitudes, the weird buzz you catch bits and pieces of and can’t quite put together…well, if there’s any unifying story, I think it’s this one.

# The State of Four Gravitons

This blog is named for a question: does the four-graviton amplitude in N=8 supergravity diverge?

Over the years, Zvi Bern and a growing cast of collaborators have been trying to answer that question. They worked their way up, loop by loop, until they stalled at five loops. Last year, they finally broke the stall, and last week, they published the result of the five-loop calculation. They find that N=8 supergravity does not diverge at five loops in four dimensions, but does diverge in 24/5 dimensions. I thought I’d write a brief FAQ about the status so far.

Q: Wait a minute, 24/5 dimensions? What does that mean? Are you talking about fractals, or…

Nothing so exotic. The number 24/5 comes from a regularization trick. When we’re calculating an amplitude that might be divergent, one way to deal with it is to treat the dimension like a free variable. You can then see what happens as you vary the dimension, and see when the amplitude starts diverging. If the dimension is an integer, then this ends up matching a more physics-based picture, where you start with a theory in eleven dimensions and curl up the extra ones until you get to the dimension you’re looking for. For fractional dimensions, it’s not clear that there’s any physical picture like this: it’s just a way to talk about how close something is to diverging.

Q: I’m really confused. What’s a graviton? What is supergravity? What’s a divergence?

I don’t have enough space to explain these things here, but that’s why I write handbooks. Here are explanations of gravitons, supersymmetry, and (N=8) supergravity, loops, and divergences. Please let me know if anything in those explanations is unclear, or if you have any more questions.

Q: Why do people think that N=8 supergravity will diverge at seven loops?

There’s a useful rule of thumb in quantum field theory: anything that can happen, will happen. In this case, that means if there’s a way for a theory to diverge that’s consistent with the symmetries of the theory, then it almost always does diverge. In the past, that meant that people expected N=8 supergravity to diverge at five loops. However, researchers found a previously unknown symmetry that looked like it would forbid the five-loop divergence, and only allow a divergence at seven loops (in four dimensions). Zvi and co.’s calculation confirms that the five-loop divergence doesn’t show up.

More generally, string theory not only avoids divergences but clears up other phenomena, like black holes. These two things seem tied together: string theory cleans up problems in quantum gravity in a consistent, unified way. There isn’t a clear way for N=8 supergravity on its own to clean up these kinds of problems, which makes some people skeptical that it can match string theory’s advantages. Either way N=8 supergravity, unlike string theory, isn’t a candidate theory of nature by itself: it would need to be modified in order to describe our world, and no-one has suggested a way to do that.

Q: Why do people think that N=8 supergravity won’t diverge at seven loops?

There’s a useful rule of thumb in amplitudes: amplitudes are weird. In studying amplitudes we often notice unexpected simplifications, patterns that uncover new principles that weren’t obvious before.

Gravity in general seems to have a lot of these kinds of simplifications. Even without any loops, its behavior is surprisingly tame: it’s a theory that we can build up piece by piece from the three-particle interaction, even though naively we shouldn’t be able to (for the experts: I’m talking about large-z behavior in BCFW). This behavior seems to have an effect on one-loop amplitudes as well. There are other ways in which gravity seems better-behaved than expected, overall this suggests that we still have a fair ways to go before we understand all of the symmetries of gravity theories.

Supersymmetric gravity in particular also seems unusually well-behaved. N=5 supergravity was expected to diverge at four loops, but doesn’t. N=4 supergravity does diverge at four loops, but that seems to be due to an effect that is specific to that case (for the experts: an anomaly).

For N=8 specifically, a suggestive hint came from varying the dimension. If you checked the dimension in which the theory diverged at each loop, you’d find it matched the divergences of another theory, N=4 super Yang-Mills. At $l$ loops, N=4 super Yang-Mills diverges in dimension $4+6/l$. From that formula, you can see that no matter how much you increase $l$, you’ll never get to four dimensions: in four dimensions, N=4 super Yang-Mills doesn’t diverge.

At five loops, N=4 super Yang-Mills diverges in 26/5 dimensions. Zvi Bern made a bet with supergravity expert Kelly Stelle that the dimension would be the same for N=8 supergravity: a bottle of California wine from Bern versus English wine from Stelle. Now that they’ve found a divergence in 24/5 dimensions instead, Stelle will likely be getting his wine soon.

Q: It sounds like the calculation was pretty tough. Can they still make it to seven loops?

I think so, yes. Doing the five-loop calculation they noticed simplifications, clever tricks uncovered by even more clever grad students. The end result is that if they just want to find out whether the theory diverges then they don’t have to do the “whole calculation”, just part of it. This simplifies things a lot. They’ll probably have to find a few more simplifications to make seven loops viable, but I’m optimistic that they’ll find them, and in the meantime the new tricks should have some applications in other theories.

Q: What do you think? Will the theory diverge?

I’m not sure.

To be honest, I’m a bit less optimistic than I used to be. The agreement of divergence dimensions between N=8 supergravity and N=4 super Yang-Mills wasn’t the strongest argument (there’s a reason why, though Stelle accepted the bet on five loops, string theorist Michael Green is waiting on seven loops for his bet). Fractional dimensions don’t obviously mean anything physically, and many of the simplifications in gravity seem specific to four dimensions. Still, it was suggestive, the kind of “motivation” that gets a conjecture started.

Without that motivation, none of the remaining arguments are specific to N=8. I still think unexpected simplifications are likely, that gravity overall behaves better than we yet appreciate. I still would bet on seven loops being finite. But I’m less confident about what it would mean for the theory overall. That’s going to take more serious analysis, digging in to the anomaly in N=4 supergravity and seeing what generalizes. It does at least seem like Zvi and co. are prepared to undertake that analysis.

Regardless, it’s still worth pushing for seven loops. Having that kind of heavy-duty calculation in our sub-field forces us to improve our mathematical technology, in the same way that space programs and particle colliders drive technology in the wider world. If you think your new amplitudes method is more efficient than the alternatives, the push to seven loops is the ideal stress test. Jacob Bourjaily likes to tell me how his prescriptive unitarity technique is better than what Zvi and co. are doing, this is our chance to find out!

Overall, I still stand by what I say in my blog’s sidebar. I’m interested in N=8 supergravity, I’d love to find out whether the four-graviton amplitude diverges…and now that the calculation is once again making progress, I expect that I will.

# At the GGI Lectures on the Theory of Fundamental Interactions

I’m at the Galileo Galilei Institute for Theoretical Physics in Florence at their winter school, the GGI Lectures on the Theory of Fundamental Interactions. Next week I’ll be helping Lance Dixon teach Amplitudeology, this week, I’m catching the tail end of Ira Rothstein’s lectures.

The Galileo Galilei Institute, at the end of a long, winding road filled with small, speedy cars and motorcycles, in classic Italian fashion

Rothstein has been heavily involved in doing gravitational wave calculations using tools from quantum field theory, something that has recently captured a lot of interest from amplitudes people. Specifically, he uses Effective Field Theory, theories that are “effectively” true at some scale but hide away higher-energy physics. In the case of gravitational waves, these theories are a powerful way to calculate the waves that LIGO and VIRGO can observe without using the full machinery of general relativity.

After seeing Rothstein’s lectures, I’m reminded of something he pointed out at the QCD Meets Gravity conference in December. He emphasized then that even if amplitudes people get very good at drawing diagrams for classical general relativity, that won’t be the whole story: there’s a series of corrections needed to “match” between the quantities LIGO is able to see and the ones we’re able to calculate. Different methods incorporate these corrections in different ways, and the most intuitive approach for us amplitudes folks may still end up cumbersome once all the corrections are included. In typical amplitudes fashion, this just makes me wonder if there’s a shortcut: some way to compute, not just a piece that gets plugged in to an Effective Field Theory story, but the waves LIGO sees in one fell swoop (or at least, the part where gravity is weak enough that our methods are still useful). That’s probably a bit naive of me, though.

# 4gravitons Meets QCD Meets Gravity

I’m at UCLA this week, for the workshop QCD Meets Gravity. I haven’t worked on QCD or gravity yet, so I’m mostly here as an interested observer, and as an excuse to enjoy Los Angeles in December.

QCD Meets Gravity is a conference centered around the various ways that “gravity is Yang-Mills squared”. There are a number of tricks that let you “square” calculations in Yang-Mills theories (a type of theory that includes QCD) to get calculations in gravity, and this conference showcased most of them.

At Amplitudes this summer, I was disappointed there were so few surprises. QCD Meets Gravity was different, with several talks on new or preliminary results, including one by Julio Parra-Martinez where the paper went up in the last few minutes of the talk! Yu-tin Huang talked about his (still-unpublished) work with Nima Arkani-Hamed on “UV/IR Polytopes”. The story there is a bit like the conformal bootstrap, with constraints (in this case based on positivity) marking off a space of “allowed” theories. String theory, interestingly, is quite close to the boundary of what is allowed. Enrico Herrmann is working on a way to figure out which gravity integrands are going to diverge without actually integrating them, while Simon Caron-Huot, in his characteristic out-of-the-box style, is wondering whether supersymmetric black holes precess. We also heard a bit more about a few recent papers. Oliver Schlotterer’s talk cleared up one thing: apparently the GEF functions he defines in his paper on one-loop “Z theory” are pronounced “Jeff”. I kept waiting for him to announce “Jeff theory”, but unfortunately no such luck. Sebastian Mizera’s talk was a very clear explanation of intersection theory, the subject of his recent paper. As it turns out, intersection theory is the study of mathematical objects like the Beta function (which shows up extensively in string theory), taking them apart in a way very reminiscent of the “squaring” story of Yang-Mills and gravity.

The heart of the workshop this year was gravitational waves. Since LIGO started running, amplitudes researchers (including, briefly, me) have been looking for ways to get involved. This conference’s goal was to bring together amplitudes people and the gravitational wave community, to get a clearer idea of what we can contribute. Between talks and discussions, I feel like we all understand the problem better. Some things that the amplitudes community thought were required, like breaking the symmetries of special relativity, turn out to be accidents of how the gravitational wave community calculates things: approximations that made things easier for them, but make things harder for us. There are areas in which we can make progress quite soon, even areas in which amplitudes people have already made progress. The detectors for which the new predictions matter might still be in the future (LIGO can measure two or three “loops”, LISA will see up to four), but they will eventually be measured. Amplitudes and gravitational wave physics could turn out to be a very fruitful partnership.

# An Elliptical Workout

I study scattering amplitudes, probabilities that particles scatter off each other.

In particular, I’ve studied them using polylogarithmic functions. Polylogarithmic functions can be taken apart into “logs”, which obey identities much like logarithms do. They’re convenient and nice, and for my favorite theory of N=4 super Yang-Mills they’re almost all you need.

Well, until ten particles get involved, anyway.

That’s when you start needing elliptic integrals, and elliptic polylogarithms. These integrals substitute one of the “logs” of a polylogarithm with an integration over an elliptic curve.

And with Jacob Bourjaily, Andrew McLeod, Marcus Spradlin, and Matthias Wilhelm, I’ve now computed one.

This one, to be specific

Our paper, The Elliptic Double-Box Integral, went up on the arXiv last night.

The last few weeks have been a frenzy of work, finishing up our calculations and writing the paper. It’s the fastest I’ve ever gotten a paper out, which has been a unique experience.

Computing this integral required new, so far unpublished tricks by Jake Bourjaily, as well as some rather powerful software and Mark Spradlin’s extensive expertise in simplifying polylogarithms. In the end, we got the integral into a “canonical” form, one other papers had proposed as the right way to represent it, with the elliptic curve in a form standardized by Weierstrass.

One of the advantages of fixing a “canonical” form is that it should make identities obvious. If two integrals are actually the same, then writing them according to the same canonical rules should make that clear. This is one of the nice things about polylogarithms, where these identities are really just identities between logs and the right form is comparatively easy to find.

Surprisingly, the form we found doesn’t do this. We can write down an integral in our “canonical” form that looks different, but really is the same as our original integral. The form other papers had suggested, while handy, can’t be the final canonical form.

What the final form should be, we don’t yet know. We have some ideas, but we’re also curious what other groups are thinking. We’re relatively new to elliptic integrals, and there are other groups with much more experience with them, some with papers coming out soon. As far as we know they’re calculating slightly different integrals, ones more relevant for the real world than for N=4 super Yang-Mills. It’s going to be interesting seeing what they come up with. So if you want to follow this topic, don’t just watch for our names on the arXiv: look for Claude Duhr and Falko Dulat, Luise Adams and Stefan Weinzierl. In the elliptic world, big things are coming.

# Interesting Work at the IAS

I’m visiting the Institute for Advanced Study this week, on the outskirts of Princeton’s impressively Gothic campus.

The IAS was designed as a place for researchers to work with minimal distraction, and we’re taking full advantage of it. (Though I wouldn’t mind a few more basic distractions…dinner closer than thirty minutes away for example.)

The amplitudes community seems to be busily working as well, with several interesting papers going up on the arXiv this week, four with some connection to the IAS.

Carlos Mafra and Oliver Schlotterer’s paper about one-loop string amplitudes mentions visiting the IAS in the acknowledgements. Mafra and Schlotterer have found a “double-copy” structure in the one-loop open string. Loosely, “double-copy” refers to situations in which one theory can be described as two theories “multiplied together”, like how “gravity is Yang-Mills squared”. Normally, open strings would be the “Yang-Mills” in that equation, with their “squares”, closed strings, giving gravity. Here though, open strings themselves are described as a “product” of two different pieces, a Yang-Mills part and one that takes care of the “stringiness”. You may remember me talking about something like this and calling it “Z theory”. That was at “tree level”, for the simplest string diagrams. This paper updates the technology to one-loop, where the part taking care of the “stringiness” has a more sophisticated mathematical structure. It’s pretty nontrivial for this kind of structure to survive at one loop, and it suggests something deeper is going on.

Yvonne Geyer (IAS) and Ricardo Monteiro (non-IAS) work on the ambitwistor string, a string theory-like setup for calculating particle physics amplitudes. Their paper shows how this setup can be used for one-loop amplitudes in a wide range of theories, in particular theories without supersymmetry. This makes some patterns that were observed before quite a bit clearer, and leads to a fairly concise way of writing the amplitudes.

Nima-watchers will be excited about a paper by Nima Arkani-Hamed and his student Yuntao Bai (IAS) and Song He and his student Gongwang Yan (non-IAS). This paper is one that has been promised for quite some time, Nima talked about it at Amplitudes last summer. Nima is famous for the amplituhedron, an abstract geometrical object that encodes amplitudes in one specific theory, N=4 super Yang-Mills. Song He is known for the Cachazo-He-Yuan (or CHY) string, a string-theory like picture of particle scattering in a very general class of theories that is closely related to the ambitwistor string. Collaborating, they’ve managed to link the two pictures together, and in doing so take the first step to generalizing the amplituhedron to other theories. In order to do this they had to think about the amplituhedron not in terms of some abstract space, but in terms of the actual momenta of the particles they’re colliding. This is important because the amplituhedron’s abstract space is very specific to N=4 super Yang-Mills, with supersymmetry in some sense built in, while momenta can be written down for any particles. Once they had mastered this trick, they could encode other things in this space of momenta: colors of quarks, for example. Using this, they’ve managed to find amplituhedron-like structure in the CHY string, and in a few particular theories. They still can’t do everything the amplituhedron can, in particular the amplituhedron can go to any number of loops while the structures they’re finding are tree-level. But the core trick they’re using looks very powerful. I’ve been hearing hints about the trick from Nima for so long that I had forgotten they hadn’t published it yet, now that they have I’m excited to see what the amplitudes community manages to do with it.

Finally, last night a paper by Igor Prlina, Marcus Spradlin, James Stankowicz, Stefan Stanojevic, and Anastasia Volovich went up while three of the authors were visiting the IAS. The paper deals with Landau equations, a method to classify and predict the singularities of amplitudes. By combining this method with the amplituhedron they’ve already made substantial progress, and this paper serves as a fairly thorough proof of principle, using the method to comprehensively catalog the singularities of one-loop amplitudes. In this case I’ve been assured that they have papers at higher loops in the works, so it will be interesting to see how powerful this method ends up being.